I have put this question on Math Overflow
Let $X$ be a projective $\mathbb{Q}$-factorial variety ("variety" is irreducible and reduced over a field of characteristic zero; not necessarily algebraically closed, but we can add this if necessary), let $\pi:X\to Y$ be a map with irreducible fibres, where $Y$ is a projective variety, and let $F$ be the reduction of a fibre of $Y$ (so $F$ is a projective variety).
My question is this:
If $C$ is a curve contained in $F$, then do we have $K_F\cdot C\leq K_X\cdot C$?
I've tried to keep the question somewhat general to see if there is a general answer, but my motivation comes from the case where $\pi$ is an elementary MMP step and $X$ is a spherical variety.
There is a combinatorial proof for the case of toric varieties and MMP; see Lemma 7.1 of "Approximating rational points on toric varieties" by McKinnon and Satriano. But this proof is very specific to the toric case because it uses the fan combinatorics. I would hope that there is a general proof of this that works for more general varieties and does not rely on combinatorics.
(Edit for clarity: $K_X$ is the canonical divisor of $X$).